2000Mathematics Subject Classification..Abstract. This is is a draft a one term course Complex Analysis for the 2nd year.
ContentsChapter 1.Geometry, topology and analysis in the complex plane11.A Review of Complex Numbers12.Geometry, topology in the Complex Plane53.Functions, their limits and continuity7Chapter 2.Derivatives and holomorphic functions111.Derivatives112.The Cauchy-Riemann Equations133.Properties of differentiable functions174.Harmonic Functions19Chapter 3.Power series and examples of holomorphic functions211.Complex series212.Power series223.The exponential and trigonometric functions254.The Logarithm Function and Branches28Chapter 4.Conformal mappings311.Paths312.Conformal mapping313.Examples of conformal maps33Chapter 5.Contour Integration and Cauchy’s Theorem371.Paths and Contours372.Path Integrals, antiderivatives383.The Cauchy-Goursat Theorem434.The Cauchy Integral Formula and its consequences48Chapter 6.Roots and singularities591.Generalised Power Series: Laurent expansion592.Proof of Theorem 6.1643.Zeros664.Counting roots and poles695.Evaluation of Real Integrals71i
CHAPTER 1Geometry, topology and analysis in the complex plane1. A Review of Complex Numbers1.1. Basic facts.Letz=(x,y) be a point on the planeR2, or, in other words, vectors onthe plane. From Linear Algebra we know how to add vectors and multiply them by real-valuedconstants:z1+z2=(x1,y1)+(x2,y2)=(x1+x2,y1+y2),az=(ax,ay),a∈R.We are going to call these vectorscomplex numbers, and name the coordinatesxandytherealandimaginary partofz:x=Rez,y=Imz.The question: why do we want these new names for the familiar objects? Answer: because weintroduce a new operation – multiplication.Definition1.1. Define the product of two complex numbersz1=(x1,y1),z2=(x2,y2) asfollows:z1z2=(x1x2-y1y2,x1y2+y1x2).Note thatz1z2=z2z1andz1(z2+z3)=z1z2+z1z3!Notation for the set of all complex numbers:C. When we view them as points on the plane,we call it theArgand plane.Consider the product of two equal numbersz1=z2=(0,1):(0,1)2=(-1,0).So, if we denotei=(0,1) andx=(x,0), then we have the familiar identityi2=-1. Now, usingthe same notation we can use another, more standard way of writing the complex numbers:z=(x,y)=x(1,0)+y(0,1)=x+iy.This is called thecanonical, or standardform of complex numbers. We are going to use thisform from now on. It is more convenient for remembering how to multiply complex numbers.The inverse numberz-1=1zis defined to be the complex number such thatzz-1=1. It iseasy to check thatz-1=x-iyx2+y2=xx2+y2-iyx2+y2.